I will provide an introductory level overview of recent
applications of resurgent trans-series and Picard-Lefschetz theory to
quantum mechanics and quantum field theory.
Resurgence connects local perturbative data with global topological
structures. In quantum mechanical systems, this program provides a
constructive relation between different saddles. For example, in
certain cases it has been shown that all information around the
instanton saddle is encoded in perturbation theory around the
perturbative saddle. In quantum field theory, such as sigma-models
compactified on a circle, neutral bions provide a semi-classical
interpretation of the elusive IR-renormalon, and fractional kink
instantons lead to the non-perturbatively induced mass gap exactly of
order of the strong scale. I also describe the concept of hidden
topological angle, a phase associated with Lefschetz thimbles.
Hidden topological angle may provide destructive/constructive
interference effects between equally dominant saddles in the Lefschetz
thimble decomposition, providing resolution to some time standing
puzzles in non-perturbative analysis.